If \(-\Delta \)  is not an eigenvalue then \(G\)  must have exactly 3 eigenvalues or exactly 2 eigenvalues. The only connected regular graphs with exactly 3 eigenvalues are the strongly regular graphs, as is show in [1]}. Moreover \(\lambda =\mu \)  is required so that one eigenvalue is the negative of the other. Finally, if \(G\)  has only two eigenvalues it must be a complete graph.
